Financial Derivatives Toolbox

hjmvolspec

Specify an HJM forward rate volatility process

Syntax

Volspec = hjmvolspec(varargin)

Arguments

Volatility Specification	Formula
Constant	sigma(t,T) = Sigma_0
Stationary	sigma(t,T) = Vol(T-t) = Vol(Term)
Exponential	sigma(t,T) = Sigma_0*exp(-Lambda * (T-t))
Vasicek, Hull-White	sigma(t,T) = Sigma_0*exp(-Decay(T-t))
Proportional	sigma(t,T) = Prop(T-t)*max(SpotRate(t),MaxSpot)

Sigma_0 is the scalar base volatility over a unit time.

Lambda is the scalar decay factor.

CurveVol is a number of curves (NCURVE) -by-1 vector of Vol values at sample points.

CurveDecay is an NCURVE-by-1 vector of Decay values at sample points.

CurveProp is an NCURVE-by-1 vector of Prop values at sample points.

CurveTerm is an NCURVE-by-1 vector of term sample points T-t.

The time values T, t, and Term are in coupon interval units specified by the Compounding input of hjmtimespec. For instance if Compounding is 2, Term = 1 is a semiannual period (six months).

Description

hjmvolspec specifies a HJM forward rate volatility process. The volatility process is sigma(t,T), where t is the observation time and T is the starting time of a forward rate. In a stationary process the volatility term is T-t. Multiple factors can be specified sequentially. Each factor is specified with one of the functional forms:

Constant volatility (Ho-Lee): VolSpec = hjmvolspec('Constant', Sigma_0)

Stationary volatility:
VolSpec = hjmvolspec('Stationary', CurveVol, CurveTerm)

Exponential volatility:
VolSpec = hjmvolspec('Exponential', Sigma_0, Lambda)

Vasicek, Hull-White:
VolSpec = hjmvolspec('Vasicek', Sigma_0, CurveDecay, CurveTerm)

Nearly proportional stationary:
VolSpec = hjmvolspec('Proportional', CurveProp, CurveTerm, MaxSpot)

VolSpec is a structure specifying the volatility model for hjmtree.

Example

Single-factor proportional

CurveProp = [0.11765; 0.08825; 0.06865];
CurveTerm = [1; 2; 3];
VolSpec = hjmvolspec('Proportional', CurveProp, CurveTerm, 1e6)

 VolSpec = 
          FinObj: 'HJMVolSpec'
    FactorModels: {'Proportional'}
      FactorArgs: {{1x3 cell}}
      SigmaShift: 0
      NumFactors: 1
       NumBranch: 2
         PBranch: [0.5000 0.5000]
     Fact2Branch: [-1 1]

Two-factor exponential and constant

VolSpec = hjmvolspec('Exponential', 0.1, 1, 'Constant', 0.2)

VolSpec = 
          FinObj: 'HJMVolSpec'
    FactorModels: {'Exponential'  'Constant'}
      FactorArgs: {{1x2 cell}  {1x1 cell}}
      SigmaShift: 0
      NumFactors: 2
       NumBranch: 3
         PBranch: [0.2500 0.2500 0.5000]
     Fact2Branch: [2x3 double]

See Also

hjmtimespec, hjmtree

hjmtree

instadd